Q1: Let the position vectors of the points P, Q, R and S be and respectively. Then which of the following statements is true?
(a) The points P, Q, R and S are NOT coplanar
(b) is the position vector of a point which divides PR internally in the ratio 5 : 4
(c) is the position vector of a point which divides PR externally in the ratio 5 : 4
(d) The square of the magnitude of the vector is 95 [JEE Advanced 2023 Paper 2]
Ans: (b)
Q2: Let ℓ1 and ℓ2 be the lines respectively. Let X be the set of all the planes H that contain the line ℓ1. For a plane H, let d(H) denote the smallest possible distance between the points of ℓ2 and H. Let H_{0} be a plane in X for which d(H_{0}) is the maximum value of d(H) as H varies over all planes in X.
Match each entry in ListI to the correct entries in ListII.
For plane
d(H)= Smallest possible distance between the points of ℓ_{2} and Plane.
d(H_{0}) = Maximum value of d(H)
For d(H_{0})
ℓ_{2} is Parallel to plane containing ℓ_{1}
Equation of plane
∵ It contain ℓ_{1}
∴ a + b + c = 0 ..........(1)
For largest possible distance between plane (1) and ℓ_{2} the line ℓ_{2} must be parallel to plane (1)
∴ Point of intersection (1, 1, 1) Distance from origin
=
Q3: Let P be the plane and let and the distance of (α, β, γ) from the plane P is 7/2. Let be three distinct vectors in S such that . Let S be the volume of the parallelepiped determined by vectors . Then the value of is : [JEE Advanced 2023 Paper 1]
Ans: 45
are elements of set S and in set S magnitude of vector is 1
∴ are unit vectors and by equation (1) we can system are equally inclined and vertices of equilateral triangle also lying on a circle which is intersection of sphere
Distance from Origin to P
Equation of the plane is
Equation of sphere =
∴ Radius or circle
∴ Area or triangle =
Velocity of Parallelepiped,
= 45
be three vectors such that and
Then, which of the following is/are TRUE?
(a)
(b)
(c)
(d) [JEE Advanced 2022 Paper 2]
Ans: (b), (c) & (d)
Given,
Adding (1), (2) and (3), we get
Aso given,
Now,
And
Comparing value of with equation (4), we get
Multiplying both side with , we get
∴ B is correct
As,
∴ Option (D) is correct.
Given,
Now,
∴ Option (C) is correct.
Q2: Let P_{1} and P_{2} be two planes given by
Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on P_{1}_{ }and P_{2} ?
(a)
(b)
(c)
(d) [JEE Advanced 2022 Paper 1]
Ans: (a), (b) & (d)
P_{1} and P_{2} be two planes given by
Now finding line of intersection of both the planes,
Let z = λ, then
Now solving the eq. (1) and (2) we get,
Now any skew line with the line of intersection of given plane can be edge of tertrahedron.
Now using above concept we will solve all options.
For option (A)
Now satisfying this point in given plane we have,
Now we can see line is intersecting the plane P_{1}, at some point.
Now checking for plane (P_{2})
Also intersecting plane (P_{2})
Hence, it can be the edge of tetrahedron.
For option (B)
this point is satisfying plane P_{1}
Now checking for plane P_{2}
Hence, it can be the edge of tetrahedron.
For option (D),
point (λ, −2λ + 4, 3λ) and for λ = 0 point will be (0, −4, 0) which is lying on line of intersection and DR of plane P_{2} is (−2,5,4) and DR of line is (1,−2,3)
Now line is lying completely on P_{2}
Hence, it can be the edge of tetrahedron.
Q3: Let S be the reflection of a point Q with respect to the plane given by
where t, p are real parameters and are the unit vectors along the three positive coordinate axes. If the position vectors of Q and S are and respectively, then which of the following is/are TRUE ?
(a)
(b)
(c)
(d) [JEE Advanced 2022 Paper 1]
Ans: (a), (b) & (c)
Given : equation of plane
on rearranging we get,
So, equation of plane in standard form is given by
Now given coordinate of Q = (10, 15, 20)
And Coordinates of
Now using the image formula of point and plane we get,
Now solving all options.
Q1: Let α, β and γ be real numbers such that the system of linear equations
x + 2y + 3z = α
4x + 5y + 6z = β
7x + 8y + 9z = γ − 1
is consistent. Let  M  represent the determinant of the matrix
Let P be the plane containing all those (α, β, γ) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.
The value of  M  is _________. [JEE Advanced 2021]
Ans: 1
On equating the coefficients,
4A + B = 7 .... (i)
5A + 2B = 8 .... (ii)
and − (γ − 1) = − Aβ − αB ..... (iii)
On solving Eqs. (i) and (ii), we get A = 2 and B = −1
From Eq. (iii), we get
Now, determinant of
[from Eq. (iv)]
Q2: Let α, β and γ be real numbers such that the system of linear equations
x + 2y + 3z = α
4x + 5y + 6z = β
7x + 8y + 9z = γ − 1
is consistent. Let  M  represent the determinant of the matrix
Let P be the plane containing all those (α, β, γ) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.
The value of  D  is _________. [JEE Advanced 2021]
Ans: 1.5
On equating the coefficients,
4A + B = 7 .... (i)
5A + 2B = 8 .... (ii)
and − (γ − 1) = − Aβ − αB ..... (iii)
On solving Eqs. (i) and (ii), we get A = 2 and B = −1
From Eq. (iii), we get
Now, determinant of
[from Eq. (iv)]
Equation of plane P is given by x −2y + z = 1
Hence, perpendicular distance of the point (0, 1, 0) from the plane
Q1: Let α^{2} + β^{2} + γ^{2 }≠ 0 and α + γ = 1. Suppose the point (3, 2, −1) is the mirror image of the point (1, 0, −1) with respect to the plane αx + βy + γz = δ. Then which of the following statements is/are TRUE?
(a) α + β = 2
(b) δ − γ = 3
(c) δ + β = 4
(d) α + β + γ = δ` [JEE Advanced 2020 Paper 2]
Ans: (a), (b) & (c)
Since, the point A(3, 2, −1) is the mirror image of the point B(1, 0, −1) with respect to the plane αx + βy + γz = δ, then
it is given that
And, the midpoint of AB, M(2, 1, −1) lies on the given plane, so
Q2: Let a and b be positive real numbers. Suppose and are adjacent sides of a parallelogram PQRS. Let u and v be the projection vectors of along PQ and PS, respectively. If u + v = w and if the area of the parallelogram PQRS is 8, then which of the following statements is/are TRUE?
(a) a + b = 4
(b) a − b = 2
(c) The length of the diagonal PR of the parallelogram PQRS is 4
(d) w is an angle bisector of the vectors PQ and PS [JEE Advanced 2020 Paper 2]
Ans: (a) & (c)
Given vectors and are adjacent sides of a parallelogram PQRS.
so area of parallelogram PQRS =
PQ × PS = 2ab = 8 (given)
∴ ab = 4 ......(i)
According to the question,
u = projection vector of along PQ
and, similarly, v = projection vector of along PS
From Eqs. (i) and (ii), we get
and the length of diagonal PR
And, the angle bisector of vector PQ and PS is along the vector
Q1: Three lines
For which point(s) Q on L_{2} can we find a point P on L_{1} and a point R on L_{3} so that P, Q and R are collinear?
(a)
(b)
(c)
(d) [JEE Advanced 2019 Paper 2]
Ans: (c) & (d)
Given lines,
Now, let the point P on L_{1} = (λ, 0, 0)
the point Q on L_{2} = (0, μ, 1), and
the point R on L_{3} = (1, 1, v)
For collinearity of points P, Q and R, there should be a nonzero scalar 'm', such that PQ = m PR
Hence, Q can not have coordinator (0, 0, 1 ) and (0, 1, 1)
Hence, options (c) and (d) are correct.
Q2: Let L_{1} and L_{2} denote the lines
respectively. If L_{3} is a line which is perpendicular to both L_{1} and L_{2} and cuts both of them, then which of the following options describe(s) L_{3}?
(a)
(b)
(c)
(d) [JEE Advanced 2019 Paper 1]
Ans: (a), (b) & (c)
Given lines,
and since line L_{3} is perpendicular to both lines L_{1} and L_{2}.
Then a vector along L_{3} will be,
Now, let a general point on line L_{1}.
and on line L_{2} as and let P and Q are point of intersection of lines L_{1}, L_{3} and L_{2}, L_{3}, so direction ratio's of L_{3}
[from Eqs. (i) and (ii)]
Now, we can take equation of line L_{3} as , where a is position vector of any point on line L_{3} and possible vector of a are
Hence, options (a), (b) and (c) are correct.
Q1: Let P_{1} : 2x + y − z = 3 and P_{2} : x + 2y + z = 2 be two planes. Then, which of the following statement(s) is(are) TRUE?
(a) The line of intersection of P1 and P2 has direction ratios 1, 2, −1
Here,
and
(a) Direction ratio of the line of intersection of P_{1} and P_{2} is
Hence, statement a is false.
(b) We have,
his line is parallel to the line of intersection of P_{1} and P_{2}.
Hence, statement (b) is false.
(c) Let acute angle between P_{1} and P_{2} be θ.
We know that,
Hence, statement (c) is true.
(d) Equation of plane passing through the point (4, 2, −2) and perpendicular to the line of intersection of P_{1} and P_{2} is
Now, distance of the point (2, 1, 1) from the plane x − y + z = 0 is
Hence, statement (d) is true.
Q2: Let P be a point in the first octant, whose image Q in the plane x + y = 3 (that is, the line segment PQ is perpendicular to the plane x + y = 3 and the midpoint of PQ lies in the plane x + y = 3) lies on the Zaxis. Let the distance of P from the Xaxis be 5. If R is the image of P in the XYplane, then the length of PR is _____. [JEE Advanced 2018 Paper 2]
Ans: 8
Let P(α, β, γ) and R is image of P in the XYplane.
∴ R(α, β, γ)
Also, Q is the image of P in the plane x + y = 3
Since, Q is lies on Zaxis
Given, distance of P from Xaxis be 5
Then,
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1. What are the key topics covered in the JEE Advanced Vector Algebra and 3D Geometry section? 
2. How important is the Vector Algebra and 3D Geometry section in the JEE Advanced exam? 
3. What are some common types of questions asked in the Vector Algebra and 3D Geometry section of JEE Advanced? 
4. How can students effectively prepare for the Vector Algebra and 3D Geometry section of JEE Advanced? 
5. Are there any specific tips or tricks to tackle challenging questions in the Vector Algebra and 3D Geometry section of JEE Advanced? 
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